Linearization Scheme of Shallow Water Equations for Quantum Algorithms

TL;DR

This paper explores a quantum algorithm-based linearization scheme for the shallow water equations, aiming to enable faster solutions for fluid dynamics problems like tsunami modeling using quantum computing.

Contribution

It extends a linearization scheme to the shallow water equations and demonstrates its potential for quantum speedup in solving fluid dynamics problems.

Findings

Quantum linear system solver performs well with the linearized equations.

The approach shows promise for exponential speedup over classical methods.

Benchmarking indicates key parameters influence quantum algorithm efficiency.

Abstract

Computational fluid dynamics lies at the heart of many issues in science and engineering, but solving the associated partial differential equations remains computationally demanding. With the rise of quantum computing, new approaches have emerged to address these challenges. In this work, we investigate the potential of quantum algorithms for solving the shallow water equations, which are, for example, used to model tsunami dynamics. By extending a linearization scheme previously developed in [Phys. Rev. Research 7, 013036 (2025)] for the Navier-Stokes equations, we create a mapping from the nonlinear shallow water equation to a linear system of equations, which, in principle, can be solved exponentially faster on a quantum device than on a classical computer. To validate our approach, we compare its results to an analytical solution and benchmark its dependence on key parameters. Additionally, we implement a quantum linear system solver based on quantum singular value transformation and study its performance in connection to our mapping. Our results demonstrate the potential of applying quantum algorithms to fluid dynamics problems and highlight necessary considerations for future developments.